1. CSc4820/6820 Computer Graphics Algorithms Ying Zhu Georgia State University
Lecture 25
Radiosity

2. Outline What is radiosity? Radiosity process
Form factor
Solve radiosity
Advantages and limitations of radiosity

3. Background
Radiosity method is based on the theory of thermal heat transfer.
Heat transfer theory describes radiation as the transfer of energy from a surface when that surface has been thermally excited.
This thermal radiation theory can be used to describe the transfer of many kinds of energy between surfaces, including light energy.

4. What is radiosity?
The radiosity of a surface is the rate at which energy leaves that surface (energy per unit time per unit area).
It includes the energy emitted by a surface as well as the energy reflected from other surfaces.
Radiosity methods allow the intensity of radiant energy arriving at a surface to be computed.
These intensities can then be used to determine the shading of the surface.
In graphics, the term “radiosity” refers to the rendering algorithm that models the lighting process as energy transfer

5. Radiosity Process Process Group mesh surface into patches
Calculate form factors for patches
Solve radiosity
Display patches

6. The Radiosity Equation
The "radiosity equation" describes the amount of energy which can be emitted from a surface, as the sum of the energy inherent in the surface (a light source, for example) and the energy which strikes the surface, being emitted from some other surface.
The energy which leaves a surface (surface "j") and strikes another surface (surface "i") is attenuated by two factors:
the "form factor" between surfaces "i" and "j", which accounts for the physical relationship between the two surfaces
the reflectivity of surface "i", which will absorb a certain percentage of light energy which strikes the surface.

7. The Radiosity Equation

8. Form Factor
The "form factor" describes the fraction of energy which leaves one surface and arrives at a second surface.
It takes into account the distance between the surfaces, computed as the distance between the center of each of the surfaces.
It also takes into account their orientation in space relative to each other, computed as the angle between each surface's normal vector and a vector drawn from the center of one surface to the center of the other surface.

9. Form Factor
To use this form factor with surfaces which have a positive area, the equation must be integrated over one or both surface areas.
The form factor between a point on one surface and another surface with positive area can be used if the assumption is made that the single point is representative of all of the points on the surface.

10. The Form Factor

11. The Nusselt Analog
Differentiation of the basic form factor equation is difficult even for simple surfaces.
Nusselt developed a geometric analog which allows the simple and accurate calculation of the form factor between a surface and a point on a second surface.
The "Nusselt analog" involves placing a hemispherical projection body, with unit radius, at a point on a surface.
The second surface is spherically projected onto the projection body, then cylindrically projected onto the base of the hemisphere.
The form factor is, then, the area projected on the base of the hemisphere divided by the area of the base of the hemisphere.

12. The Nusselt Analog

13. The Hemicube Approximation
In the "hemicube" form factor calculation, the center of a cube is placed at a point on a surface
The upper half of the cube is used as a projection body as defined by the "Nusselt analog."
Each face of the hemicube is subdivided into a set of small, usually square ("discrete") areas, each of which has a pre-computed form factor value.
When a surface is projected onto the hemicube, the sum of the form factor values of the discrete areas of the hemicube faces which are covered by the projection of the surface is the form factor between the point on the first surface (about which the cube is placed) and the second surface (the one which was projected).

14. The Hemicube Approximation
The speed and accuracy of this method of form factor calculation can be affected by changing the size and number of discrete areas on the faces of the hemicube.

15. The Hemicube Anolog
This illustration demonstrates the calculation of form factors between a particular surface on the wall of a room and several surfaces of objects in the room.

16. The Hemicube Anolog
A radiosity algorithm will compute the form factors from a point on a surface to all other surfaces, by projecting all other surfaces onto the hemicube and storing, at each discrete area, the identifying index of the surface that is closest to the point.
When all surfaces have been projected onto the hemicube, the discrete areas contain the indices of the surfaces which are ultimately visible to the point.
From there the form factors between the point and the surfaces are calculated.
For greater accuracy, a large surface would typically be broken into a set of small surfaces before any form factor calculation is performed.

17. Radiosity form factor learniing tools
Brown University Exploratories project:
Radiosity Form Factor:
This applet shows how the form factor between two patches varies in magnitude depending on the relative geometry of the patches.

18. The Full Matrix Radiosity Algorithm
Two classes of radiosity algorithms have been developed which will calculate the energy equilibrium solution in an environment.
The "full matrix" radiosity solution calculates the form factors between each pair of surfaces in the environment, then forms a series of simultaneous linear equations.
This matrix equation is solved for the "B" values, which can be used as the final intensity (or color) value of each surface.

19. The Full Matrix Radiosity Algorithm

20. The Full Matrix Radiosity Algorithm
This method produces a complete solution, at the substantial cost of first calculating form factors between each pair of surfaces and then the solution of the matrix equation.
Each of these steps can be quite expensive if the number of surfaces is large
complex environments typically have upwards of ten thousand surfaces, and environments with one million surfaces are not uncommon.
This leads to substantial costs not only in computation time but in storage.

21. The Progressive Radiosity Algorithm
The "progressive" radiosity solution is an incremental method, yielding intermediate results at much lower computation and storage costs.
Each iteration of the algorithm requires the calculation of form factors between a point on a single surface and all other surfaces, rather than all N-squared form factors (where "N" is the number of surfaces in the environment).
After the form factor calculation, radiosity values for the surfaces of the environment are updated.

22. The Progressive Radiosity Algorithm
This method will eventually produce the same complete solution as the "full matrix" method, though, unlike the "full matrix" method, it will also produce intermediate results, each more accurate than the last.
It can be halted when the desired approximation is reached.
It also exacts no large (again, N-squared) storage cost.

23. The Progressive Radiosity Algorithm

24. Progressive Radiosity Example

25. Progressive Radiosity Variants
Several variations on the basic progressive radiosity algorithm have been developed.
“Gathering” variant
“Shooting” variant
The "gathering" variant updates one surface by collecting light energy from all other surfaces.
In this variant, as well as the "shooting" variant, the "base" surface is arbitrarily chosen.

26. Progressive Radiosity Variants
The "shooting" variant updates all surfaces by distributing light energy from one surface
The "shooting and sorting" variant chooses the surface with the greatest unshot light energy and distributes its light energy to the surfaces in the environment.
The "shooting and sorting" method is the most desirable, as it finds the surface with the greatest potential contribution to the intensity solution and updates all other surfaces in the environment with its energy.
In addition to these, an initial "ambient" term can be approximated for the environment and adjusted at each iteration, gradually replaced by the true ambient contribution to the rendered image.

27. Radiosity Learning Tool
Brown University Exploratories project:
Radiosity Shooting vs Gathering
This applet shows the graphic difference between two radiosity algorithms: shooting and gathering.
The algorithms are animated for each iteration of the algorithms and the resulting visualization of a discretized scene and of its form factor matrix can be viewed as the animation progresses.

28. Comparison of Progressive Variants
shooting
gathering
Shooting & sorting & ambient
Shooting & sorting

29. The Two-Pass Radiosity Solution
View independent, global diffuse illumination computed with radiosity pre-process.
View dependent, global specular illumination computed with ray tracing post-process.
Combining the strengths of radiosity and ray tracing achieves a more accurate and efficient solution to the problem of light transport.

30. Participating Media
Another variation on the basic diffuse radiosity solution adds the contribution of light passing through a participating medium, such as smoke, fog, or water vapor in the air
Creating volumetric lighting effect
In this algorithm, light energy is sent through a three-dimensional volume representing a participating medium, which both attenuates the light energy and adds to the intensity solution through illumination of the participating medium.

31. Participating Media

32. Advantages of Radiosity
Photorealistic image quality
Accurate simulation of energy transfer
Can simulate color bleeding.
Light reflected from a surface is attenuated by the reflectivity of the surface, which is closely associated with the color of the surface.
The reflected light energy often is colored, to some small extent, by the color of the surface from which it was reflected.

33. Advantages of Radiosity
The reflection of light energy in an environment often produces a phenomenon known as "color bleeding," where a brightly colored surface's color will "bleed" onto adjacent surfaces. The image in this slide illustrates this phenomenon, as both the red and blue walls "bleed" their color onto the white walls, ceiling and floor.

34. Radiosity vs. Ray Tracing
Simulate soft shadows: shadows with soft edges
Ray tracing often creates hard-edged shadows
Viewpoint independent: the solution will be the same regardless of the viewpoint of the image
Ray tracing is viewpoint dependent
Radiosity is an object space algorithm while ray tracing is an image space algorithm

35. Limitations of Radiosity
Large computational and storage costs
Must preprocess polygonal environments
Difficult to compute form factor
Does not consider transparent/translucent surfaces
Non-diffuse components of light not represented
Specular highlights hard to achieve

36. Radiosity Examples

37. Radiosity Examples
Over 1 million polygons

38. Radiosity Examples

39. Radiosity tools Pixar’s PRMan Radiosity is supported in POV-Ray

40. Summary What is radiosity? Radiosity process
Form factor
Solve radiosity
Advantages and limitations of radiosity

41. Readings HyperGraph: Radiosity Brown University Exploratories project
Brown University Exploratories project

42. Next lecture
Animation